He is a very patient teacher with a very sympathic voice and charisma.
@sophisticatedplayer
6 ай бұрын
10:55 The top part was a perfect square, you don't even need to distribute everything ((t + 1) + (t - 1))^2 = (2t + 1 - 1)^2 = (2t)^2 = 4t^2
@PrimeNewtons
6 ай бұрын
Haha! Now I see it.
@tiramisu_1th
6 ай бұрын
yup, this is the comment im looking for
@Mycroft616
5 ай бұрын
That is how I handled it, too.
@LucenProject
5 ай бұрын
Yup, came for this!
@Zachs_Facts
2 ай бұрын
I noticed that and was wondering whether you would use it.
@VictorGarcia-gv1ri
5 ай бұрын
I love you're enthusiasm. It makes me feel like I'm not crazy or left alone because sometimes I find math or science fascinating and when I try to talk to people about it they look at me weird. We need more teachers like you.
@stevenwilson5556
4 ай бұрын
Agreed, he has a perfect attitude to teach!
@paulw176
5 ай бұрын
hey, I'm 65 and just starting to do some math again. I was able to follow that long forgotten algebra so thanks, that is encouraging - subscribed.
@yamada.masahiro
6 ай бұрын
Your way of solving it is universal. Great! I found the numerator of RHS equals ( x + 2 )^2, and then I tried to express the denominator with ( x + 2) and ( x - 2 ). 8 x = ( x + 2 )^2 - ( x - 2 )^2 ∴ RHS = ( x + 2 )^2 ÷ { ( x + 2 )^2 - ( x - 2 )^2 } = { ( x + 2 ) / ( x - 2 ) }^2 ÷ [ { ( x + 2 ) / ( x - 2 ) }^2 - 1 ] Replace ( x + 2 ) / ( x - 2 ) with x, you can get x^2 / ( x^2 - 1 )
@shaswatadutta4451
5 ай бұрын
I did exactly the same thing!!
@PooshanHalder
4 ай бұрын
Same thing I did
@metadivergence9523
3 ай бұрын
Me too
@shashankks8758
Ай бұрын
I tried same thing but missed in expressing 8x in terms of X+2 ad X-2 , thanks for the steps
@heywrandom8924
Ай бұрын
I also did this but the identity for the denominator might not be known by many so I will try make a story that might help finding solutions in the future. We will try to guess the function. First notice the x square in the numerator which means that there is some squaring involved. So try f(t) = t^2. You get the numerator of the Right Hand Side (RHS) but not the denominator. You can multiply and divide the denominator by the thing you want which is (x-2)^2. Then you have 8x/(x-2)^2 in the denominator. Issue is that there is no evident simplification unless you saw the relevant identity in the past and remembered it. So we will have to write 8x in some way that involves (x-2)^2. If in the end we want to write a function of (x+2)/(x-2) we will probably need to write 8x in terms of (x-2)^2 and (x+2). If we want to get rid of the x^2 in (x-2)^2 when that term is expanded then it might be interesting to look at (x-2)^2-(x+2)^2.
@malefetsanekoalane4549
2 ай бұрын
I am a pensioner and I alternate between doing math and the garden.Your presentation is just so captivating. I just can't imagine what I would be doing if I couldn't do math .Kudos from Johannesburg. Been thinking that functional equations were reserved for IMOs. 😅
@PrimeNewtons
Ай бұрын
So nice of you
@boguslawszostak1784
4 ай бұрын
I prefer a clear and simple formulation to avoid any confusion In the first and second lines, the letter 'x' is used in different ways. We're used to writing y=f(x), so it's easier to change the 'x' to 't' in the first line. This gives us the equations: f(x)=y x=(t+2)/(t-2) y=(t^2+4t+4)/(8t) Our task is to eliminate the variable 't' from these equations. (t-2)*x=(t+2) x*t-2x=t+2 x*t-t=2x+2 t=2(x+1)/(x-1) y=(t+2)^2/(8t)= ... etc
@MervynRThomas
2 ай бұрын
I love the fun you have with maths. Your enthusiasm is infectious. I wish my teachers had had half your ability.
@joydeepdas4735
6 ай бұрын
12:38 you can simply write the numerator as [ (t+1)+(t-1) ]^2=(2t)^2=4t^2
@Hardman7
3 ай бұрын
Never stop teaching Coach ! Thanks
@matthewkendall5235
6 ай бұрын
Neat algebra - you might wish to explain how the original function won't given an answer at x = 2, whereas the revised function won't give an answer at x = 1 or -1 and how that works okay - as you have shifted the points where the function doesn't converge because of a divide by zero and why that would be allowed!
@jacobgoldman5780
6 ай бұрын
You should specify that x cannot be 0 or 2 in domain of f as those values are not in the domain of original functional equation.
@glorrin
6 ай бұрын
that's not entirely correct the final x is not the same as the first one. t cannot be 1 or -1. and if f(x) = t^2/(t^2-1) then x cannot be 1 or -1. But t = x+2/(x-2), then whatever x t cannot be 1 so there is no problem here. t = -1 when x = 0 so you have one exception in common x = 0 is the same as t = -1. when x = 2 t is not defined so there is no problem. The first equation is not defined on 0 and -2 but the answer is not defined on 1 and -1
@PrimeNewtons
6 ай бұрын
I was just looking for f(x).
@JuniperHatesTwitterlikeHandles
6 ай бұрын
in the original statement f((x+2)/(x-2)) = (x+2)^2/8x you would _not_ input 2 into that function by replacing the x with 2, because x is not the input to the function. You would replace the x with a number such that (x+2)/(x-2) is equal to 2, because (x+2)/(x-2) is the input to the function. (6+2)/(6-2)=2 (6+2)^2)/8(6)=4/3 thus 2 is in the domain of the original function, you can watch him work out how 0 is in the domain of the original function in the beginning of the video. Changing the value you put into a function does not change the function or its domain. If we had a separate function g, defined so that g(x) = f((x+2)/(x-2)) then _that_ function, g, would not be defined at 2, but f still is, because when you feed 2 into f, it returns 4/3.
@adw1z
6 ай бұрын
Yes u are almost right (I see what u were trying to say) - clearly plugging in x = 0 ==> there is a simple pole at t = -1 for f(t) and taking some limit e.g. let x -> 2+ ==> f(t) -> 1 as t-> +inf let x -> 2- ==> f(t) -> 1 as t -> -inf This can be seen all from the initial question (and clearly holds with the final answer!), but all he wanted to do was find the function, which he did - not specify the domain and range of the functional equation (which is an obvious 2 second job anybody can do). Slight mistake in your comment: the domain of f(x+2/x-2) has those problems, not the domain of f itself; domain of f only has a singularity at -1
@UgyenRangdol-gf8cc
6 ай бұрын
Every one in the comment going crazy
@chrispatrikios9823
3 ай бұрын
That quote at the end sent me. Very enjoyable personality.
@mrunalhatekar11
6 ай бұрын
Your content is so good that i think you deserve atleast a million subs. I am from India and i love watching your content. If for any reason you get depressed or think that you should stop making your videos, there's always me and my group of friends watching your vdos. Your teaching skills are fabulous. The way you make maths interesting. Thanks a lot my man. Love from india
@PrimeNewtons
6 ай бұрын
Wow! That means a lot to me. Thank you, and God bless.
@zk513
2 ай бұрын
Functional equations were always very cruel to me. Thanks to you, I'm starting to see the light. Keep on teaching!
@IRanOutOfPhrases
5 ай бұрын
Been WAAAAY too long since I looked at this stuff. I was always pretty good and keen on math, but once this stuff started to turn up, it made the subject loads more interesting. It's hard to describe, but the way these functions relate to one another, it almost feels like you're peeling away at the layers of how the universe as a whole operates. Some of the discoveries end up being more exciting than others, of course. Very similar vibes with how taking the derivative of a function, and then taking that derivative, and then taking that derivative, and all these functions you end up with all relate to one another. It's like the numbers behind the numbers behind the numbers. You're introduced to things like parabolas and other common graph shapes well before learning derivatives, so it just felt like a huge plot twist when you first learn that these derivatives were there 'driving' the shapes of the graph all along. I don't know, just always seemed very cool to me.
@favourtube527
5 ай бұрын
I am from Bangladesh. And mymother langyage is not english. But your lecture is incredible. Despite being a bangali i can understand your solution so easily.your way of teaching is not boring at all. You are a really great teacher
@mdforhad-wk1zo
2 ай бұрын
Same bro🇧🇩🇧🇩
@gooddude9211
4 ай бұрын
Excellent sir. Loved the way you simplified and great explanation.
@Rhiannon_Autumn
3 ай бұрын
that's amazing. Never seen functional equations before but solving that looked like a lot of fun.
@karryy02
5 ай бұрын
The solution is actually easy. On the first sight, we can already see that 8x = (x+2)² - (x-2)², let u = (x+2)/(x-2), the eq becomes f(u)=1/(1-u⁻²) And that's the function we need to find.
@holmbrg-_-2221
6 ай бұрын
Great videos you make, they are super useful. For me personally i have, in the last couple of days, learned a bunch of new techniques from your videos.
@beaumatthews6411
2 ай бұрын
Hey, really nice. I noticed something though, before 12:36 but at that time it's the step above the one you're pointing at. The top is the form a^2 + 2ab + b^2, so it equals (a+b)^2, which is ((t+1)+(t-1))^2 which evaluates to (2t)^2 then 4t^2, which is what you end up with as well. Just thought it was interesting, I immediately noticed it when I saw it
@labbertubes
5 ай бұрын
this is one of the most compelling math videos it has been my joy to behold. Nice cap, too.
@tubetigeerr
5 ай бұрын
i really like the syle he talks/teaches here!!
@sounakkundu6115
6 ай бұрын
As x approaches 2 from 2+ or 2- we see that the value is 1, thus allowing us to find f(t) as t approaches both negative and positive infinity. Mind Blown.
@marxcarton3858
5 ай бұрын
This is where i learnt how to solve functional equations, thank you so much!!
@nihadsaid286
4 ай бұрын
انا من فلسطين . واحب الرياضيات . انت مذهل و رائع . ساتابعك باستمرار . تحياتي
@subarnodatta
5 ай бұрын
Sir I am an Indian student studying in class 12th (high school).. i substituted t = x+2/x-2, and then directly used COMPONENDO-DIVIDENDO to get t+1/t-1 = x/2.. so x = 2(t+1/(t-1)).. then I directly considered (x^2 + 4x + 4)/8x as (x+2)^2/8x and substituted x as 2(t+1/(t-1)) on both the sides to get the desired answer. Thanks a lot for this question sir..
@freddyalvaradamaranon304
5 ай бұрын
Muy interesante, didáctica y buena clase, a mi hija le servirá mucho esta excelente exposición. Estamos muy agradecidos con su bella persona, bendiciones y éxitos para Usted y su linda familia. ❤
@EDWING6017
5 ай бұрын
Excellent, very interesting this exercise. Thanks so much!!! Greeting from Perú!
@alangrant5278
2 ай бұрын
I’m very happy to have found your channel!!!
@YakobWakjera
23 күн бұрын
I love your introduction sir...
@johnstebbins6262
5 ай бұрын
Wonderfully clear explanation!
@Sal_da_Terra21
5 ай бұрын
Nice lesson! Congratulations teacher.
@grinreaperoftrolls7528
6 ай бұрын
THIS IS SO COOL! How was this never taught to me? I made it all the way through cal 3 without ever knowing this
@fisimath40
6 ай бұрын
You have very good content and scenic mastery. The form presented shows the equivalence with the change of variable It could also have been done like this x²+4x+4=(x+2)² (x²+4x+4)/8x=(x+2)²/8x, dividing numerator and denominator by (x-2)² =((x+2)²/(x-2)²)/(8x/(x-2)²), adding and subtracting 1 from the denominator =((x+2)/(x-2))²/(8x/(x-2)²+1-1) =((x+2)/(x-2))²/(((x+2)/(x-2))²-1) then the change f(x)=x²/(x²-1)
@brahimaboumohamed1284
5 ай бұрын
Yes, Cooool !!!!
@tanoukouassialban7501
Ай бұрын
Very nice. I like your videos. Just continue
@grandstrategy8987
5 ай бұрын
easy to understand. you're a great teacher!
@glebannya
6 ай бұрын
Bro you are great! I'm studying maths profoundly at school and your content is exactly what I'm obsessed with. Thank you!
@user-qx6br9ds2p
5 ай бұрын
Your presentation is awesome.
@BlackPhoton
5 ай бұрын
Great channel, I really appreciate what you're doing and how you explain math concepts. Regarding this algebra the only thing I miss is to determine the function domain which is also part of the solution.
@benhuralexandredossantos1996
2 ай бұрын
Sou muito fã de suas aulas! Obrigado muito!
@anatoliy3323
5 ай бұрын
Thanks a lot for your interesting lesson, sir! 👍💯
@franklinokafor
24 күн бұрын
Keep up the good work Sir!❤ From Nigeria😊
@thexavier666
5 ай бұрын
Your enthusiasm is very nice
@john-paulderosa7217
5 ай бұрын
Wonderful manner that conveys such enthusiasm and positivity. I would have understood better if a graph of the function had been included when it was found. That might have helped understand the domain issues that got so many commenters in knots.
@GargamelTheGreat
2 ай бұрын
We need more math teachers like this dude
@jpete190
3 ай бұрын
Thank you, awesome training.
@parthtandel7443
3 ай бұрын
Hey I don’t understand at the end…at first you took x+2/x-2 = t…so it gives you X= 2t + 2/ t-1…but at the end why did you implied t = X
@fannyliem3536
2 ай бұрын
It's a month late, but basically since f(t) means that t is a variable of that function, t could be replaced by other variable including x. However t ≠ x, like you said, t = x+2/x-2, t is not equal to x. And f(t) ≠ f(x). Basically means that t is not implied to be equal to x, the variable that is used in the function is changed, not made equal. For example if f(x) = x+2 Let's say x = 2 and t = 3 Then f(t) = t+2 = 3+2 = 5 And f(x) = x+2 = 2+2 = 4 the formula of (x+2) is still the same, but x ≠ t because 2 ≠ 3 and f(t) ≠ f(x) I hope anyone reading this comment understood my explanation.
@3v4battler
Күн бұрын
@@fannyliem3536i got your point but aren’t we supposed to calculate the actual f(x)’s value? It is like saying i found the f(P) but not f(x) but since i can rename the variable let’s replace P by x…..the value which we got at last is not of the actual defined function.
@michaelbaum6796
6 ай бұрын
Very nice equation 👍
@andreasretsinas2142
4 ай бұрын
Very nice video! Students will love it! Keep going!
@roronoazoro8343
5 ай бұрын
i like this person man, such a happy intraction
@kailasnathastro
5 ай бұрын
The way you explain the steps and logic is really remarkable and I enjoy all your videos.
@puckchibi
3 ай бұрын
You are a very good teacher!
@Scienceguy0
4 ай бұрын
The best Math teacher i have ever seen Iam from egypt And iam a new subscriber YOU MAKE MATH FUN🎉 THX❤❤❤❤
@ajamessssss
6 ай бұрын
Excellent video sir, i thoroughly enjoyed it. just by looking at the thumbnail. I guessed we would have to plug in another variable, But I made the mistake of substituting a directly into the equation. like, f(a) = (((x+1)/(x-1))^2 + 4(x+1)/(x-1) + 4 )/ 8((x+1)/(x-1))
@mdforhad-wk1zo
2 ай бұрын
Dream math teacher around the world❤❤❤
@naumtrandos4191
5 ай бұрын
A mathematics video has never had a harder plot twist than this 🔥
@vivekrajput..
5 ай бұрын
You have a Amazing attitude A god's gift
@edmurnico7508
5 ай бұрын
Very, very nice explanation! Greetings from Brasil
@johnnyfarfan1723
3 ай бұрын
Excelente ejercicio. Saludos.
@ritwikgupta3655
25 күн бұрын
You have great "board-side" manner. Cool...But sometimes shorter methods are easier to follow. Put x+2=a, x-2=b and a/b=c, then, f((x+2)/(x-2)) is f(a/b) or f(c) and RHS = a^2/(a^2-b^2) = 1/(1-(b/a)^2) = 1/(1-(1/c)^2) = c^2/(c^2-1) Now, as f(c)=c^2/(c^2-1) Substituting x for c, gives f(x)= x^2/(x^2-1)
@nYEOSUh
5 ай бұрын
t로 치환하는 방법은 미처 몰랐네요. 멋진 아이디어 감사합니다!
@Jperre2010
6 ай бұрын
Excellent blackboard techniques.
@matematicasyotrostemas
5 ай бұрын
Nice explanation.
@paulg444
5 ай бұрын
Great video !!
@v.purushothamanvedachalam737
5 ай бұрын
Excellent, sir
@calculotv1
5 ай бұрын
Excellent!!
@MohammedAli-jt7zr
5 ай бұрын
honestly i liked your explanation quite a lot dam it was interesting how you explained great respect from India Ali 🖖👍
@elophostia
3 ай бұрын
You are awesome, subscribed immediately!
@DasWan-qs6sx
Ай бұрын
Very nice video !
@ChaosPod
6 ай бұрын
10:49 You could have factorised the numerator (t+1)^2 + 2(t+1)(t-1) + (t-1)^2 = ((t+1) + (t-1))^2 = (2t)^2 = 4t^2 since it is of the form (a + b)^2 = a^2 +2ab +b^2
@jandirpassos5327
5 ай бұрын
Very good. Greetings from Brazil
@saarike
2 ай бұрын
Wonderful!!!
@mihaipuiu6231
6 ай бұрын
Nice....nice.
@gourivigneshgembali3496
5 ай бұрын
I have a very easy solution. in the RHS, the numerator can be written as (x+2) ^2 and denominator can be written as ((x+2) ^2 - (x-2) ^2) and then divide the numerator and denominator with (x-2) ^2. Then replace x + 2/x - 2 with x. The solution is x^2/x^2 - 1
@francescocastaldi4045
16 күн бұрын
Why did you not substitute t with (x+2)/(x-2) at the end?
@enzostecher
4 ай бұрын
Nice! 👏👏👏👏
@rezaghajar6564
2 күн бұрын
13:18 Why can you switch the f(t) = (t^2)/( (t^2) - 1) to f(x) = (x^2)/( (x^2) - 1) when we defined t as (x + 2) / (x - 1)? It doesn't make sense to me because I think that is like saying t = x, when at the start it did not. Was the goal just to get the input to be one letter?
@biscuit_6081
5 ай бұрын
The change of variables from calc 2 at the end is so nice
@peterc.hayward8067
2 ай бұрын
This is the part I didn't understand! Why can you arbitrarily decide to call it x again? I thought x was defined in a specific way
@user-sh6xv5yj6m
5 ай бұрын
FELICIDADES ERES MUY BUENO
@rogerfroud300
5 ай бұрын
I hated maths at school, yet here I am watching this and enjoying it now I'm retired. I guess we just didn't have very good teachers.
@jonathanestrada1064
5 ай бұрын
So my takeaway is that when given a functional equation call it f(g(x)) in order to determine f(x) we simply find the inverse of g(x) so that when we plug that into f(g(x)) we get f(x). Sounds simple enough! Very good example I just wish he would have mentioned the technique in more general terms at the end. After all as a mathematician we want to be able to generalize results.
@PrimeNewtons
5 ай бұрын
What you explained is brilliant. That wasn't my strategy in any way. I would try that next time. Thanks
@stevenwilson5556
4 ай бұрын
I never studied functional equations and I have a degree in math with 4 semesters of calculus under my belt. I did not focus on algebra, more on probability and statistics and this sort of mathematics does not come up much in that area of math.
@vincentkobani-rn5zh
5 ай бұрын
I am inspired by you my Brother
@simeon7450
Ай бұрын
Why can you replace t with x at the end? I understand replacing t with another variable like z or y. But t had already been defined in terms of x (t=(x+2)(x-2)); so replacing t with x directly (t=x) seems weird to me because you are overwriting the previous assumption?
@mcichael9661
Ай бұрын
Bob Ross of algebra
@kenbrashear210
2 ай бұрын
Well done.
@henry_dschu
5 ай бұрын
yeah, this is what we did in ms. the method is that which is called the substitution of variates. make t = (x+2)/(x-2)(x≠2 &),then t= g(x), then integrate g(x) into the function on the right side, we will get a f(t)=t²/t²-1(t≠1, x≠0)。so we have f(x)=x²/x²-1(x≠±1 & x≠0 & x≠2)
@tubesteaknyouri
5 ай бұрын
Thank you. You are like the Bob Ross of math.
@nanamacapagal8342
5 ай бұрын
I love your solution! I mostly just winged it, tried x = 1 and x = -1 for f(5) and f(3), then picked more values of x for f(-3) and f(-5) Then I checked by plugging back in (x+2)/(x-2)
@WepWip
5 ай бұрын
👍 great
@3v4battler
Күн бұрын
Aren’t you supposed to put the value of “t” i.e. (x+2)/(x-2) in the last step cause just by simply replacing the the letter t by x won’t give you the actual f(x) do let me know if I’m incorrect.
@skwbusaidi
15 күн бұрын
That big niominator can be simplify without expanding by noticing that it is a prefect square (t+1)^2 +2(t+1)(t-1) + (t-1)^2 =(t+1+t-1)^2 =(2t)^2 =4t^2
@kyoushlro
Ай бұрын
In school I learned this approach was called the dummy variable approach.
@plante2166
6 ай бұрын
This is how I did it: Notice that x^2+4x+4 = (x+2)^2. Substitute u=x+2 to get f(u/(u-4)) = u^2/(8(u-2)). So the idea would be to express u^2/(8(u-2)) in terms of u/(u-4). The first thing to try would be squaring u/(u-4), which gives us u^2/(u^2-8u+16) = u^2/(u^2-8(u-2)). We note that this looks very similar to the right hand side already, save for the extra u^2 in the denominator. This inspires us to try 1/(u/(u-4))^2 = (u^2-8(u-2))/(u^2) = 1-8(u-2)/(u^2) which is just 1-f(u/(u-4)). So now we have 1/X^2 = 1-f(X) (writing u/(u-4) as X). We solve for f to obtain f(X) = X^2/(X^2-1).
@RobertSmith-lu2rm
5 ай бұрын
Excellent
@borisdelaine9797
6 ай бұрын
This brings back great memories. Do dat math!
@ThePROestRedn99
2 ай бұрын
Did in mind in 2 mins.....by just dividing by x-2 whole square and then manupulating the terms😊
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